projecting onto a mirror and back?
As has been said, Cycles isn't capable of this.  Luxrender, however, is.  You can read about it on this forum and elsewhere.  However, I'll say here that it integrates into Blender (previewing is even possible with the experimental API enabled as demonstrated below), and it handles refractive and reflective caustics very well.  It is a bit slower but the versatility is worth it for much more accurate physics than Cycles.

I have reproduced your scene by creating a Projector Spot Lamp (using your image) and a Mirror material on the reflector.  The reflector mesh has Subsurf and has Smooth Shading enabled.  The "screen" has the default Matte material:



Here I am morphing the plane from flat, to convex, to concave:



You must install Luxrender before opening the .blend: